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590 Chapter 10 ROUND-OFF EFFECTS IN DIGITAL FILTERS
10.4 PROBLEMS
P10.1 Let x(n) =0.5[cos(n/17) + sin(n/23)]. For the following parts, use 500, 000 samples of
x(n) and the StatModelR function.
1. Quantize x(n) toB =2bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
2. Quantize x(n) toB =4bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
3. Quantize x(n) toB =6bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
P10.2 Let x(n) = 1 [cos(0.1πn)+sin(0.2πn)+sin(0.4πn)]. For the following parts use 500, 000
3
samples of x(n) and the StatModelR function.
1. Quantize x(n) toB =2bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
2. Quantize x(n) toB =4bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
3. Quantize x(n) toB =6bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
P10.3 Let a real, causal, and stable IIR filter be given by
N−1
∑ R k
H(z) =R 0 +
(10.105)
z − p k
where all poles are distinct. Using (10.16), (10.18a), and (10.105), show that
k=1
σq
2
σe
2
N−1 N−1
∑ ∑
= R0 2 R k Rl
∗ +
1 − p k p ∗ l
k=1
l=1
P10.4 Consider the lowpass digital filter designed in Problem P6.39. The input to this filter is
an independent and identically distributed Gaussian sequence with zero-mean and
variance equal to 0.1.
1. Determine the variance of the filter output process using the VarGain function.
2. Determine numerically the variance of the output process by generating 500,000
samples of the input sequence. Comment on your results.
P10.5 Design an elliptic bandpass digital filter that has a lower stopband of 0.3π, alower
passband of 0.4π, anupper passband of 0.5π, and an upper stopband of 0.65π. The
passband ripple is 0.1 dB and the stopband attenuation is 50 dB. The input signal is a
random sequence whose components are independent and uniformly distributed between
−1 and 1.
1. Determine the variance of the filter output process using the VarGain function.
2. Determine numerically the variance of the output process by generating 500,000
samples of the input sequence. Comment on your results.
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